Two-sample hypothesis testing-determining whether two sets of data are drawn from the same distribution-is a fundamental problem in statistics and machine learning with broad scientific applications. In the context of nonparametric testing, maximum mean discrepancy (MMD) has gained popularity as a test statistic due to its flexibility and strong theoretical foundations. However, its use in large-scale scenarios is plagued by high computational costs. In this work, we use a Nystr\"om approximation of the MMD to design a computationally efficient and practical testing algorithm while preserving statistical guarantees. Our main result is a finite-sample bound on the power of the proposed test for distributions that are sufficiently separated with respect to the MMD. The derived separation rate matches the known minimax optimal rate in this setting. We support our findings with a series of numerical experiments, emphasizing realistic scientific data.

We present the results of the "Fast Calorimeter Simulation Challenge 2022" -- the CaloChallenge. We study state-of-the-art generative models on four calorimeter shower datasets of increasing dimensionality, ranging from a few hundred voxels to a few tens of thousand voxels. The 31 individual submissions span a wide range of current popular generative architectures, including Variational AutoEncoders (VAEs), Generative Adversarial Networks (GANs), Normalizing Flows, Diffusion models, and models based on Conditional Flow Matching. We compare all submissions in terms of quality of generated calorimeter showers, as well as shower generation time and model size. To assess the quality we use a broad range of different metrics including differences in 1-dimensional histograms of observables, KPD/FPD scores, AUCs of binary classifiers, and the log-posterior of a multiclass classifier. The results of the CaloChallenge provide the most complete and comprehensive survey of cutting-edge approaches to calorimeter fast simulation to date. In addition, our work provides a uniquely detailed perspective on the important problem of how to evaluate generative models. As such, the results presented here should be applicable for other domains that use generative AI and require fast and faithful generation of samples in a large phase space.

We propose a robust methodology to evaluate the performance and computational efficiency of non-parametric two-sample tests, specifically designed for high-dimensional generative models in scientific applications such as in particle physics. The study focuses on tests built from univariate integral probability measures: the sliced Wasserstein distance and the mean of the Kolmogorov-Smirnov statistics, already discussed in the literature, and the novel sliced Kolmogorov-Smirnov statistic. These metrics can be evaluated in parallel, allowing for fast and reliable estimates of their distribution under the null hypothesis. We also compare these metrics with the recently proposed unbiased Fr\'echet Gaussian Distance and the unbiased quadratic Maximum Mean Discrepancy, computed with a quartic polynomial kernel. We evaluate the proposed tests on various distributions, focusing on their sensitivity to deformations parameterized by a single parameter $\epsilon$. Our experiments include correlated Gaussians and mixtures of Gaussians in 5, 20, and 100 dimensions, and a particle physics dataset of gluon jets from the JetNet dataset, considering both jet- and particle-level features. Our results demonstrate that one-dimensional-based tests provide a level of sensitivity comparable to other multivariate metrics, but with significantly lower computational cost, making them ideal for evaluating generative models in high-dimensional settings. This methodology offers an efficient, standardized tool for model comparison and can serve as a benchmark for more advanced tests, including machine-learning-based approaches.

The Neyman-Pearson strategy for hypothesis testing can be employed for goodness of fit if the alternative hypothesis $\rm H_1$ is generic enough not to introduce a significant bias while at the same time avoiding overfitting. A practical implementation of this idea (dubbed NPLM) has been developed in the context of high energy physics, targeting the detection in collider data of new physical effects not foreseen by the Standard Model. In this paper we initiate a comparison of this methodology with other approaches to goodness of fit, and in particular with classifier-based strategies that share strong similarities with NPLM. NPLM emerges from our comparison as more sensitive to small departures of the data from the expected distribution and not biased towards detecting specific types of anomalies while being blind to others. These features make it more suited for agnostic searches for new physics at collider experiments. Its deployment in other contexts should be investigated.

Modern high-energy physics experiments operating at colliders are extremely sophisticated devices consisting of millions of sensors sampled every few nanoseconds, producing an enormous throughput of complex data. Several types of technologies are employed, devoted to identifying and measuring the particles that originated in the collisions; in all cases, the environmental conditions are severe, making the required performances challenging to achieve. Although the various subsystems are designed to offer redundancy, measurements can be undermined by malfunctions of parts of the experiment, either because of critical inefficiencies or because of possibly misinterpreted spurious signals. In addition to supervising the status (powering, electronic configuration, temperature, etc.) of the various hardware components, data from all sources must thus be monitored continuously to assess their quality and to promptly detect any faults, possibly providing indications about their causes. Given the rate of tens of MHz at which data is gathered and the number of sensors to be checked, the monitoring process needs to be as automated as possible: approaches based on Machine Learning (ML) techniques are particularly suited for this task and have started being employed by the experimental collaborations [1-4], complementing more traditional methods [5-9].

Normalizing Flows have emerged as a powerful brand of generative models, as they not only allow for efficient sampling of complicated target distributions, but also deliver density estimation by construction. We propose here an in-depth comparison of coupling and autoregressive flows, both of the affine and rational quadratic spline type, considering four different architectures: Real-valued Non-Volume Preserving (RealNVP), Masked Autoregressive Flow (MAF), Coupling Rational Quadratic Spline (C-RQS), and Autoregressive Rational Quadratic Spline (A-RQS). We focus on different target distributions of increasing complexity with dimensionality ranging from 4 to 1000. The performances are discussed in terms of different figures of merit: the one-dimensional Wasserstein distance, the one-dimensional Kolmogorov-Smirnov test, the Frobenius norm of the difference between correlation matrices, and the training time. Our results indicate that the A-RQS algorithm stands out both in terms of accuracy and training speed. Nonetheless, all the algorithms are generally able, without much fine-tuning, to learn complex distributions with limited training data and in a reasonable time, of the order of hours on a Tesla V100 GPU. The only exception is the C-RQS, which takes significantly longer to train, and does not always provide good accuracy. All algorithms have been implemented using TensorFlow2 and TensorFlow Probability and made available on GitHub.

Precision measurements and new physics searches at the Large Hadron Collider require efficient simulations of particle propagation and interactions within the detectors. The most computationally expensive simulations involve calorimeter showers. Advances in deep generative modelling - particularly in the realm of high-dimensional data - have opened the possibility of generating realistic calorimeter showers orders of magnitude more quickly than physics-based simulation. However, the high-dimensional representation of showers belies the relative simplicity and structure of the underlying physical laws. This phenomenon is yet another example of the manifold hypothesis from machine learning, which states that high-dimensional data is supported on low-dimensional manifolds. We thus propose modelling calorimeter showers first by learning their manifold structure, and then estimating the density of data across this manifold. Learning manifold structure reduces the dimensionality of the data, which enables fast training and generation when compared with competing methods.